Madrid, Spain, currently Kassel, Germany

amparo.gil@uam.es

jsegura@math.uc3m.es

A globally convergent fixed point method
can be defined for functions satisfying second order ODEs
y_{n}^{''}+B(x)y_{n}^{'}+A_{n}y_{n}=0
(being *n* the order) and verifying linear
formulas, with continuous coefficients, relating its derivative with the same
function
and a function of a contiguous order. Theses conditions are met by a broad
family
of special functions (solutions of Bessel, Coulomb, Hermite, Legendre and
Laguerre
equations, among others) and leads to efficient algorithms for the
computation of the zeros of general solutions of these equations. We show that
similar
techniques can be applied for the computation of turning points.
The interlacing between the
zeros of a solution of a second order ODE
y_{n}^{''}+A_{n}y_{n}=0
and its derivatives when A_{n}>0 (at most, there can be
only a zero and there can be two turning points
when A_{n}0) and the monotonicity of the logarithmic derivative
y_{n}^{'}/y_{n} is the
starting point in this case. Computer algebra systems help in automatically
performing the
transformations required prior to the application of the fixed point methods.